This work proposes a new copula class that we call the MGB2 copula. The new copula originates from extracting the dependence function of the multivariate GB2 distribution (MGB2) whose marginals follow the univariate generalized beta distribution of the second kind (GB2). The MGB2 copula can capture non-elliptical and asymmetric dependencies among marginal coordinates and provides a simple formulation for multidimensional applications. The new class features positive tail dependence in the upper tail and tail independence in the lower tail. Furthermore, it includes some well-known copula classes, such as the Gaussian copula, as special or limiting cases. The validation of the MGB2 copula can be assessed by a graphical tool of the so-called “conditional plots”. To illustrate the usefulness of the MGB2 copula in practice, we build a trivariate model to analyze a data set that contains rich information on bodily injury liability claims closed within two-week period in years 1987, 1992, and 1997. Reparametrized log-F (EGB2) distributions are chosen to accommodate the right-skewness and the long-tailedness of the outcome variables,while continuous predictors are fitted by non-linear curves in the marginal regression models. The pairwise dependence structures exhibited motivate the application of the MGB2 copula. For comparison purposes we also consider the alternative Gumbel copula and t copula for the adaption of the upper tail dependence. The quantitative and graphical assessment for goodness-of-fit demonstrates the comparative advantage of the MGB2 copula over the other two copulas, which practically establishes the necessity for the development of this new copula class.